Completely positive approximation property - Imported Knowlpedia Content

Definition

A $C^*$ -algebra $A$ has the completely positive approximation property (CPAP) if there are positive integers $n(\lambda)$ and completely positive contractions

A\xrightarrow{\ \phi_\lambda\ }M_{n(\lambda)}(\mathbb C) \xrightarrow{\ \psi_\lambda\ }A

such that $\|\psi_\lambda\phi_\lambda(a)-a\|\to0$ for every $a\in A$ . The index $\lambda$ ranges over a net, so neither separability nor a sequential approximation is assumed. The convergence is point-norm: it is uniform on each fixed finite set after choosing a sufficiently advanced approximation, not uniform on the unit ball.

Finite-set formulation

Equivalently, for every finite set $F\subseteq A$ and every $\varepsilon>0$ , there are an integer $n$ and completely positive contractions $\phi:A\to M_n(\mathbb C)$ and $\psi:M_n(\mathbb C)\to A$ such that

\|\psi\phi(a)-a\|<\varepsilon\qquad(a\in F).

The matrix algebra may be replaced by a finite-dimensional $C^*$ -algebra without changing the property. This formulation displays the local content: each finite portion of $A$ is approximated through finite-dimensional operator-algebraic data while retaining positivity at every matrix level.

Equivalence with nuclearity

A $C^*$ -algebra has CPAP if and only if it is a nuclear $C^*$ -algebra Brown–Ozawa, Theorem 2.3.8. One direction uses completely positive factorizations to compare tensor norms; the other extracts finite-dimensional approximations from nuclearity. This equivalence is why CPAP is often used as the working definition of nuclearity, although the two knowls emphasize different mechanisms.

Examples and consequences

For $M_k(\mathbb C)$ , take $n=k$ and both maps equal to the identity. Finite direct sums of matrix algebras therefore have CPAP. Approximation by finite-dimensional subalgebras shows that AF algebras have CPAP, while commutative $C^*$ -algebras obtain it from partitions of unity and finite-dimensional sampling constructions.

Since each composite $\psi_\lambda\phi_\lambda$ is a finite-rank contraction, CPAP implies the metric approximation property of the underlying Banach space. The converse fails: complete positivity and the matrix-factorization structure contain information absent from ordinary finite-rank approximation Choi–Effros, pp. 61–79.

Conventions and scope

Some sources formulate CPAP using finite-rank completely positive contractions on $A$ rather than writing the two maps. For nuclearity, the factorized form above is the standard robust formulation and makes the finite-dimensional intermediate algebra explicit.

References

Man-Duen Choi and Edward G. Effros, “Nuclear C-Algebras and the Approximation Property,” American Journal of Mathematics 100 (1978), 61–79. DOI record. Relevant: completely positive finite-dimensional approximation of nuclear C-algebras.
Nathanial P. Brown and Narutaka Ozawa, C-Algebras and Finite-Dimensional Approximations*, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. DOI record. Relevant: §2.3, especially Theorem 2.3.8, on CPAP and nuclearity.